Numerical Solutions 



When the results were tested by comparing the computed ship sections with the 

 original, it was found that the agreement was excellent for all but one of 20 sec- 

 tions, but was very bad, showing double points and poor agreement at the ends, 

 in the one case of failure. The reasons for this failure will be discussed in a 

 following section. 



Another method of conformal mapping, which has been studied by many in- 

 vestigators, is that of the Gershgorin integral equation. This method is thor- 

 oughly treated by Gaier [3] from both the theoretical and practical point of view. 

 Nevertheless, other investigators have found [4] that odd- shaped forms can be 

 successfully mapped by means of the Gershgorin equation only if extreme care 

 is taken in expressing the integrals by quadrature formulas. 



The purpose of this note is to present our experience and recommendations 

 for mapping pathological double ship sections, i.e., sections with inflection 

 points and corners at the free surface and keel. Since it is intended to use the 

 resulting mathematical representation in integral equations for potential flow 

 about ship forms, economy of numerical evaluation is an important consideration. 



BIEBERBACH METHOD 

 Let 



r ^1 ^3 as /-v 



^ = ^ ^ T ^ :^ ^ :^ ^ • • • ■ ^^^ 



b 1 b -3 b r 



£=,,^,^,_1, ... , (2) 



z z 



be a transformation and its inverse which map a double ship section in the com- 

 plex z plane into a circle about the origin in the ^ plane. Here, b.^, 33 , • • • 

 and bj, b3 , • • • are real and only the coefficients with odd indices appear be- 

 cause of the symmetry of the section about the vertical and horizontal axes. 

 The Bieberbach method is based on the property that, among the closed curves 

 in the ^ plane obtained from the given section in the z plane by the transforma- 

 tion (2) for various values of bi, b3, • * •, the circle will bound the maximum 

 area. Thus, if the series in (2) is truncated, and the condition of maximizing 

 the area is applied to each of the b's, one obtains a set of linear equations for 

 determining them (Ritz procedure), as is elaborated in [Ij. Finally, Eq. (2) is 

 inverted to yield (1), since it is usually the a's that are of interest. 



Since both (1) and (2) are infinite series, we must be concerned with their 

 convergence. For (1) we can state that the series converges in the exterior of 

 the unit circle and gives a one-to-one mapping of the given profile into the unit 

 circle. For (2), however, we can only say that the series in (2) converges in the 

 exterior of the smallest circle in the z plane which circumscribes the given 

 profile. Actually, the radius of the inner circle of convergence may be reduced 

 until the radius of the singularity of the mapping function closest to the circum- 

 scribing circle. Thus, (2) will give a one-to-one convergent transformation of 

 the profile into the unit circle if and only if no singularities of the transforma- 

 tion (2) lie between the inscribed and circumscribed circles. 



1620 



